Complex Type 4 Structure Changing Dynamics of Digital Agents: Nash Equilibria of a Game with Arms Race in Innovations

Abstract

Powers of recursion are often adduced for ‘recombinant innovation’. The onset of the digital economy, based on operations on discretely encoded information, has renewed interest in this. Digital computing has a provenance with the John von Neumann dynamical systems theory based on complex biological systems. How can digital agents innovate? The Gödel-Turing-Post (GTP) logic is shown to be necessary to generate innovation based structure changing Type 4 dynamics of the Wolfram-Chomsky schema. GTP logic specifies the only known syntactic procedures that allow digital agents to initiate protean behaviours that mark exit strategies from known listable sets of digital technologies to produce novelty and surprises. The first is metaanalyses or offline simulations. The second is a fixed point which entails a two place encoding of negation or opposition, typically referred to as the Gödel sentence. It is postulated that in phenomena ranging from the genome to human proteanism, the Gödel sentence is a ubiquitous syntactic construction without which escape from hostile agents qua the Liar is not possible and digital agents become entrained within fixed repertoires. The only recursive best response function of a two person adversarial game that can implement strategic innovation in a lock-step formation of an arms race is the productive function of the Emil Post (1944) set theoretic proof of the Gödel incompleteness result. This also overturns the customary view of game theorists that surprise and innovation cannot be a Nash equilibrium of a game.